## Abstract

The theory of impact between nano-sized particles leads to the important non-dimensional grouping *γ*/*aB*, where *γ*, *a* and *B* are the effective surface energy, effective near neighbour distance, and effective bulk modulus for the impacting surfaces. A general, though formal, expression for *γ*/*aB* is obtained for a planar surface, and a modified Lennard-Jones analysis suggests that this grouping is of the order of several per cent. Bounds are suggested on this ratio, which are verified for many of the elements about their melting points, for several liquids at 298 K, and for the liquified noble gases. It is suggested that, at the atomic scale, the ratio between yield stress and the bulk modulus is also approximately equal to this grouping, which would make impacts between nano-scaled surfaces very much harder than for macroscopic impacts. However, *γ*/*aB* is material dependent and can vary by a factor of two or more about its median value. Subject to such variation, it is suggested that, to leading order, collisions between pure nanoscale particles depend primarily on the number of atoms in the clusters, and on a non-dimensional velocity scale, for low-velocity impacts. The expression for *γ*/*aB* is evaluated approximately in planar geometry by assuming that interactions between surface atoms (or molecules) only involve radial pair surface interactions.

## 1. Introduction

One of the fundamental properties of condensed matter is the existence of a surface and the associated surface energy (surface tension). Measurement of surface energy for liquids is relatively easy, but the corresponding measurements for solids are more challenging and are more difficult to define, because they become facet dependent. Surface energies often become of increasing importance as the scale of the system reduces, and at the nano scale, surface energy can become a dominant factor in system behaviour.

Adhesion effects become important for impacting nano-scale particles, a topic of technological importance (Reichel *et al*. 2006), and it is then probable that the ratio of adhesive to elastic energy becomes an important parameter, which introduces the ratio between surface energy, *γ*, and the bulk modulus, *B*. From a classical viewpoint, the ratio *γ*/*B* is apparently an arbitrary, essentially unconstrained, quantity, because surface energies are related to detailed surface asperities, which make the actual area of contact not equal to the apparent area of contact.

However, at the nanoscale and below, much of the detailed complexities of structures may be missing, since there may not be the volume of material needed for their construction. Then the possibility arises that there may be a general relationship between *γ* and *B*. Note that the ratio *γ*/*B* has dimensions of length, and for metals, *γ* is of the order of unity (N m^{−1}), and *B* of the order of 10^{10}–10^{11} N m^{−2}, so that this implicit length scale is of the order of angstroms, or less. Consequently, we expect surface energies to become important at the atomic scale, and that new phenomena may arise from the interaction between *γ* and *B*.

From the point of view of molecular dynamics, where an energy potential may be used to track the behaviour of component atoms or molecules in a nanoparticle, there is a potential link between *γ* and *B*. Specifically, the tendency to separate two surfaces, or to compress two surfaces, may be, in some sense, proportional, especially if most of the variation of the potential can be considered to occur about some minimum of uniform shape.

The aim of this paper is to develop a simple mathematical model to investigate this idea, to suggest the possible relationship between *γ* and *B*, and to attempt to check this relationship using published values of *γ* and *B*. Having demonstrated the possible existence of such a relationship in some special cases, some implications of this relationship are discussed.

## 2. General theory

In this section we consider a planar body, above which is a corresponding shell of material, one atom thick. We assume that the force *F* between the body and the shell is a function of the separation distance *x* between the body and the shell, and that a corresponding potential *Φ* can be defined as(2.1)

We proceed by assuming that this force between the body and the shell is repulsive for small distances, and attractive for large distances, and that there is a unique bounded distance of separation, *a*, for which(2.2)and we choose the constant of integration implied in (2.2) so that(2.3)

In order to separate the shell from its equilibrium position, an equal and opposite force needs to be applied to that in (2.1), and the corresponding energy per unit area to separate the shell is then(2.4)where *γ* is the surface energy, and d*A* is some reference area. The surface energy *γ* is typically positive, because the potential at *a* is typically negative.

Instead, if the shell is compressed towards its corresponding body, about its equilibrium position, by an infinitesimal distance *δx*, it encounters a repulsive force, then within the Harmonic approximation, the corresponding repulsive stress is(2.5)and so the corresponding elastic constant *B* implied in (2.5) is(2.6)Here we assume that *B* is the bulk modulus and that we are considering the same reference area d*A* in both (2.4) and (2.5), and so from (2.4) and (2.6), we have(2.7)Thus the ratio of *γ*/*aB* is defined by (2.7) and (2.2), once the potential *Φ* in (2.2) is known. Note that both sides of (2.7) are non-dimensional and invariant to any constant scaling of either *Φ* or *x*. Note also that the function *Φ* is quite general, and will be facet dependent, and will contain contributions from interactions between all of the components of the system. Thus *Φ* is essentially incalculable.

Before considering an approximate evaluation of *Φ* in §3, we consider the example of(2.8)with *m*>*n*>1. From (2.2) and (2.8), the equilibrium position for this example is(2.9)and so from (2.7)(2.10)In Lennard-Jones type potentials (Awasthi *et al*. 2007), it is usual for the exponents *m* and *n* in (2.8) to be quite large, which suggests that the ratio of *γ*/*aB* could be as low as a few per cent. Alternatively, an exponential form of repulsion is often used (Moskvich & Buznik 1979).

## 3. Idealized planar model

Section 2 provided a general expression for *γ*/*aB*, involving in principle all interparticle interactions, but this did not link directly to molecular interactions. The aim of this section is to approximate the potential *Φ* in (2.2) to intermolecular pair interactions, and to assume that only surface molecules effectively contribute to the force between the surface and the hypothetic shell discussed above. Throughout we shall consider bodies composed of the same material, so that locally, physical properties are isotropic on the surface, and we shall ignore forces in the plane.

Consider two basic entities, such as an atom or molecule, each residing on the surface of a small element of area, one of which lies in the hypothetical shell, and the other in the planar surface. Let the force (aligned along the line connecting the two surface elements) between these two entities be denoted by *f*(*R*), where *R* is the distance between the two surface elements. Then the total infinitesimal force d*F*_{12} between two elements of area, d*A*_{1} and d*A*_{2}, is(3.1)where *σ*_{1} and *σ*_{2} are the surface densities of entity 1 and entity 2 on the respective surface elements d*A*_{1} and d*A*_{2}, provided the forces from each entity are essentially independent, allowing the point-to-point forces to be simply added together to give the total force.

In general *f*(*R*) will depend on the relative orientation of the two surfaces, their separation, and on the composition of the two bodies whose surface is being discussed. If the two surface elements are widely separated, relative to the individual atomic spacings in each body, then perhaps the dominant term in *f*(*R*) will be radially symmetric, and we will assume this is the case in what follows. However, as the two surface elements approach each other, the higher order multipole terms will become important, and this assumption will begin to fail. We expect the two major contributions to *f*(*R*) to be the long-range attractive van der Waals forces, and the short-range repulsive terms associated with quantum mechanical exclusion.

We now consider a planar surface, and a small element of area, d*A*_{2}, lying at distance *x* above the plane and parallel to the planar surface. Immediately below d*A*_{2} we form a cylindrical coordinate system in the plane, with radial coordinate *y*. Then(3.2)and from (3.1), the total force *F* acting on surface element d*A*_{2} from all of the plane is vertical,(3.3)where the pairwise force *f* has a corresponding potential *ϕ*,(3.4)and again we choose the integration constant so that(3.5)

From (3.3), the total force between the plane and the surface element d*A*_{2} is proportional to *ϕ*, and so *F* can only be zero when *ϕ* is, and hence the bounded equilibrium position *a* in (2.2) is given by(3.6)Therefore we expect *a* to be smaller than the equilibrium position in the pair–pair force *f* (i.e. the molecular spacing), and so surface elements are compressed relative to intermolecular spacings, in order for the distant attractive contributions to be balanced by a local repulsive contribution.

The potential *Φ*(*a*) is then, from (2.1) and (3.3),(3.7)and so from (2.7),(3.8)

For example, if the pairwise potential *ϕ* is of Lennard-Jones form,(3.9)with *m*>*n*>2, then from (2.2) and (3.9), the equilibrium position from (3.6) is at *a*=1, and from (2.8), (3.8) and (3.9),(3.10)This expression corresponds to only nearest neighbour interactions, explaining its difference from that in (2.10).

The corresponding molecular distance, where *f* is zero, is given by the expression on the r.h.s. of (2.9). For *m*=12 and *n*=6, the molecular spacing is approximately 12% greater than that at the surface. We can estimate *a* from(3.11)where *M*_{1} is the mass of one molecule and *ρ* is the bulk density. Consequently, knowing *a* and *B*, along with estimates for *m* and *n*, will provide an (order of magnitude) estimate for *γ*, in either the liquid or the solid state.

We can now estimate the primary or limiting yield stress *Y*_{0} for our solid, applying at the unit cell scale, since the work done to move one unit cell by one lattice spacing away from a surface will equal approximately all of the work to move a unit cell to spatial infinity, which equals approximately *γa*^{2}. On the other hand, the work done by the corresponding yield of the surface will be *Y*_{0}*a*^{3}, if the primary yield stress *Y*_{0} for the solid is a measure of the work done per unit volume of displaced material. Then we have from (3.10) that(3.12)

If we assume that *m*=2*n*, and that *n* (the attractive electronic multipole contribution) varies between 3 and 6, then from (3.10), *γ*/*aB* should vary between approximately 0.025 and 0.25.

## 4. Experimental support

The suggestion in §3 that(4.1)is now tested for elements at their melting points, where data for both *γ* and *B* are readily available. Figure 1 plots the ratio *γ*/*aB* for some of the elements, at their melting points and for atmospheric pressure. The corresponding values of surface tension and density for all of the (non-noble gas) elements are tabulated in Kaye & Laby (1995). The value assumed for the nearest neighbour distance is the average value derived from (3.11). The corresponding values of bulk modulus were obtained by extrapolation to the melting points of values given in Landolt–Bornstein for the solid elastic constants, assuming that , for cubic materials. These extrapolated values will have an uncertainty of approximately 20%, because they differ by this amount from estimates for the bulk modulus (which should be continuous at the phase boundary) obtained using published data from sound speed measurements in the liquid region.

For example, *B* at the melting point can be estimated to an accuracy of over 5% (due to the accuracy of measuring sound speeds) from using liquid conditions for temperatures just above the melting temperature (Hixson *et al*. 1990), or solid conditions just below the melting temperature (Landolt & Bornstein 1979). When this is done for nickel, for example, estimates from the liquid region of approximately 80 GPa typically are less than those from the solid region (over 100 GPa); similarly, the values for iron also show the same ordering. It appears that considerable accuracy in estimates for the (isothermal) bulk modulus can be achieved for a given sample, but that significant variations occur between different samples and across phase boundaries.

It can be seen that the limits suggested for *γ*/*aB* in (4.1) hold for all of the examples in figure 1. While the values for the lattice spacing, or the near neighbour distance, do not alter much for the various elements in figure 1, there can be up to about a factor of 1000 in the variation of the values of surface tension for the different entries in figure 1. In principle, all of the solid or liquid elements could be entered in figure 1, using information from the internet, for Young's modulus, which could be equated to the bulk modulus, but unfortunately the author was unable to verify the sources of many internet entries, explaining the restricted entries in figure 1.

Figure 2 plots the ratio *γ*/*aB* for some common liquids at room temperature and pressure. These values are available in the *CRC handbook* (Beyer 2000). It is remarkable, but perhaps not unexpected, that water has the most outstanding ratio of all of the entries in figure 2, appearing to behave perhaps somewhat as a solid, and that seemingly quite different substances, such as mercury and methanol, have such similar ratios. While this is again a limited set of examples, it is interesting to note that the bounds suggested in (4.1) remain valid for this dataset.

Our final examples are the cryogenic liquified gases shown in table 1. Here the densities are from Eslami & Azin (2003), the speed of sound data are from the *CRC handbook*, and the bulk moduli have been obtained by assuming that 5*B*=3*ρc*^{2}. The value of *γ* for helium is given in the *CRC handbook*, and the other values of *γ* are from Summ (1999), whose measurements are for ‘several degrees above the melting point’. Again the ratio of *γ*/*aB* is bounded as in (4.1).

## 5. Limiting strength of materials

The limiting yield strength *Y*_{F} of a solid is given approximately by Frenkel's formula (Kittel 1967)(5.1)where *G* is the shear modulus of the material. This value of yield stress appears to be an upper bound to measured values of yield stress, which are typically hundreds or even thousands of times smaller than those given by (5.1).

The yield stress *Y* of a material is typically related to the aspect ratio of the yielding volume, because the element of work done d*W* in compressing a surface by a normal distance d*x* over an area *A* is then d*W*=*YA* d*x*=*Y* d*V*, and so the yield stress for compressional plasticity can be defined as *Y*=d*W*/d*V*, or the rate at which work is needed to be performed to plastically deform a given volume of material. This clearly depends on the geometry of the yielding surface and explains the weakening of yield stress with sample size. We use this volumetric approach here, because if a constant pressure holds about a section of surface, then plasticity can still be discussed, even for length scales below those of a grain boundary.

The shear modulus *G* and isothermal bulk modulus *B* are related to Young's modulus *E* and Poisson's ratio *ν* through(5.2)so that for *ν* about one-third, (5.2) predicts that *G* should be slightly smaller than one-third of *E*, and that *B* should approximately equal *E*. For many substances away from the melting point, *G* does indeed roughly equal somewhat less than one-third of *E*, but there are several notable exceptions where *B* does not equal *E*. For example, at 20°C and a pressure of one bar, chromium has *E*=279.1 GPa and *B*=160.1 GPa, in which case *B* is significantly smaller than *E*. Alternatively, both gold (*E*=78 GPa, *B*=217 GPa) and lead (*E*=16.1 GPa, *B*=45.8 GPa) have their bulk modulus significantly greater than that of their Young's modulus.

The two estimates for the atomic strength of materials, given in (3.12), (5.1) and (5.2), with *ν*=1/3, suggest similar values of limiting strength of materials, even though the mechanisms for providing the strength are quite different. Specifically, , which lies within the range suggested in (4.1). This suggests that *γ*/*aB* is largely a function of Poisson's ratio.

## 6. Discussion and conclusions

A theory for the ratio *γ*/*aB* was derived in (2.7), and this result was evaluated approximately for planar surfaces. The theory predicted that this ratio should vary between approximately 1/40 and 1/4, and experimental evidence was provided for this from the melting point of some of the elements, for some liquids at room temperature and for the condensed noble gases. Nevertheless, data on both surface tension and bulk modulus are quite limited, and the comparisons made in figures 1 and 2 simply reflect this.

Surface energies, at absolute zero temperature, from density functional theory (DFT) and other numerical methods are usually given to three or four significant figures (Vitos *et al*. 1998), implying that the knowledge of surface energies have great accuracy at absolute zero temperature. However, even in this limiting case, it is clear that there is considerable uncertainty in our knowledge of surface energies, in some cases. For example, the present estimates for the surface energy of lead for any facet appear to vary (Vitos *et al*. 1998) approximately (0.3–0.6) J m^{−2}.

The theory was developed by assuming that matter is continuously distributed within the bodies, and that thermal effects can be ignored in the evaluation of surface tension, which corresponds to assuming the temperature as 0 K. Temperature is important, however, as both *γ* and *B* tend to zero as the critical temperature is approached. Similarly, as the condensed state fails at the boiling temperature, *T*_{b}, we expect that surface energy and kinetic energy are of the same order, . Thus in the condensed state, the kinetic energy terms should always be less than the corresponding surface energy terms. At the melting temperature, *T*_{m}, the role of kinetic energy, while somewhat smaller than the surface energy terms, has become important, as shown in figure 3, which plots the ratio of *γa*^{2} to the thermal energy *kT*_{m}, where *k* is Boltzmann's constant. It can be seen that this ratio is approximately 3 for several materials.

A major prediction of this theory has been the contraction that should occur about solid surfaces, relative to that for molecular spacings. Specifically, the intermolecular spacing occurs at the minimum of the intermolecular potential function, whereas from (3.6), the nearest neighbour spacing about the surface of a plane occurs at the zero in the intermolecular function. For a Lennard-Jones potential, the ratio between the planar and molecular spacings is given in (2.9), which suggests the effect is of the order of 10% for many materials. This effect is easily understood because the long-range attractive interactions need to be balanced by near neighbour repulsion, requiring that layer spacings about planar surfaces are smaller than that for the corresponding molecular spacings.

The majority of research at present on surface tension *γ* concerns its dependence on temperature, or on composition. The temperature dependence of *γ* for many liquids (Adamson & Gust 1997) is described by , where *k* is Boltzmann's constant, *T* is the temperature in Kelvin, and *T*_{c} is the critical temperature. This expression for surface tension needs modification about the critical point (Ghatee *et al*. 2003) where *γ* decreases to zero as (*T*_{c}−*T*)^{1.26}. In colloid suspensions, the relationship has been derived (Marr & Gast 1994). Work on the compositional dependence of *γ* often draws on the approximately additive property of the parachor (Escobedo & Mansoori 1996) and leads to the expression that *γ* varies as (*ρ*_{l}−*ρ*_{v})^{4}, where *ρ*_{l} and *ρ*_{v} are the liquid and vapour densities. A review of the basic thermodynamics of *γ* is found in Lyklema (2001), who emphasized the wide variability of surface excess energy per unit area *U*_{a} (over three orders of magnitude), whereas the entropy per unit area *S*_{a} is almost constant for many liquids, with *S*_{a}≃0.1 mJ m^{−2} K^{−1} and . In contrast, this paper has essentially related *γ* to the bulk modulus.

Researchers have found related results, with Lord Rayleigh pointing out (Davis & Scriven 1976) that (in correcting a result of Young)(6.1)and an essentially equivalent result has been derived by Davis & Scriven (1976), except that the ratio 3/20 in (6.1) was replaced by 1/8. The term on the right of (6.1) is in Adamson & Gust (1997), where *α* is the coefficient of thermal expansion, and *T* is temperature. This is then similar to the results in (4.1) if *P*/*B* is small, and if *αT* is essentially of order unity. (Note that *αT* is unity for a perfect gas.) We have been unable to introduce the role of pressure with our approach, although it is clear that correction terms of the order of *P*/*B* should appear in (4.1).

Many aspects of the mechanics of impacts between micro- to macro-scale metal particles are now well understood, with good agreement between the theoretically derived and experimentally measured values of Newton's coefficient of restitution with respect to velocity (Johnson 1985) and also for its shape- and time dependence (Weir & Tallon 2005). Typically, the role of surface adhesion is unimportant (Johnson 1985) in these impacts, with the dominant factors being the initial kinetic energy, plastic work and elastic recoil resulting from material elasticity. Typical length scales are microns, and time units are microseconds.

Consider an idealized impact between two identical nanoparticles, characterized by a bulk modulus *B*, hardness *Y*, surface energy *γ*, typical radius *R* and initial kinetic energy . If the effects of facets can be ignored, then the impact should be primarily characterized by the four energy scales(6.2)(6.3)and so the system should be determined, to first order, by the three non-dimensional numbers _{m}/_{B}, _{Y}/_{B} and _{γ}/_{B}. The second ratio is just *Y*/*B*, which from (3.12) is about the ratio *γ*/*aB* and so will be largely insensitive for many materials. Similarly, the third ratio equals *γ*/*RB*=*γ*/*aB*×*a*/*R*, and so this ratio is simply this largely insensitive parameter of *γ*/*aB* divided by approximately the cube root of the number of molecules or atoms in the particle, which may be largely insensitive for different materials. Hence the behaviour of identical impacting nanoparticles may depend primarily on the material-dependent energy ratio of _{m}/_{B} and the number of particles in the impacting particles.

The ratio *γ*/*aB* represents a crude balance between the spacing (*a*) of atoms in solids or in liquids, and the ability of those materials to compress (*B*) or to be separated (*γ*). The term *Ba*^{3} represents a crude estimate for the energy to compress a solid to a spacing of *a*, and the term *γa*^{2} represents a crude estimate for the energy to separate this material. We can expect then that equilibrium will hold when these two terms are roughly equal, which leads to the importance of the ratio *γ*/*aB*. Recently Wang *et al*. (2006) have also identified this ratio as important in materials science and pointed out that this ratio tends to be small. An achievement of this paper is to derive (3.10), which explains why *γ*/*aB* is a small ratio.

## Acknowledgments

The author is grateful to Peter McGavin for help with figures 1–3, and to Dr Shaun Hendy for an internal review of this manuscript.

## Footnotes

- Received December 6, 2007.
- Accepted March 18, 2008.

- © 2008 The Royal Society